Convergence rate of EM scheme for SDDEs (Q2845471)

The authors investigate the existence and uniqueness of a class of continuous and pure jump processes evolving the following stochastic delay differential equations (SDDEs) and further, they study the rate of convergence of the associated Euler-Maruyama (EM) scheme. The first class of processes is driven by an \(\mathbb R^n\)-valued Brownian motion \((W_t)_{t \in [0,T]}\), \(T>0\), and defined by NEWLINE\[NEWLINEdX(t) = b(X(t),X(t-\tau)) dt + \sigma(X(t),X(t-\tau)) dW(t), \quad t \in [0,T], \tag{1}NEWLINE\]NEWLINE with initial data \(X(\theta) = \xi(\theta)\), \(\theta \in [-\tau,0]\). The second one is defined by NEWLINE\[NEWLINEdX(t) = b(X(t),X(t-\tau)) dt + \int_{U} h(X(s),X(s-\tau),u) \tilde N(ds,du), \quad t\in [0,T], \tag{2}NEWLINE\]NEWLINE with initial data \(X(\theta) = \xi(\theta)\), \(\theta \in [-\tau,0]\). The set \(U\) is a Borel set on \(\mathbb R\) and \(\tilde N(ds,du) = N(ds,du)-ds\lambda(du)\), where \(N(ds,du)\) is the Poisson counting measure and \(\lambda\) a \(\sigma\)-finite measure defined on the Borel \(\sigma\)-algebra on \(\mathbb R\).NEWLINENEWLINEThe authors show the existence and the uniqueness of a global strong solution of (1) under the assumptions that \(b\) and \(\sigma\) are Lipschitz with respect to the first argument and possibly highly nonlinear (including functions of polynomial growth of any order) with respect to the delay variable. This result is proved through Hölder, Burkholder-Davis-Gandy, Young, Gronwall inequalities used to show that the process \((X_t)_{t \in [0,T]}\) is uniformly bounded w.r.t the \(L^p\)-norm, for \(p \geq 2\).NEWLINENEWLINEFurthermore, under the previous assumptions on \(b\) and \(\sigma\) they prove that the Euler-Maruyama process resulting from the discretization of (1) using the continuous EM scheme converges strongly (uniformly on \([0,T]\) w.r.t the \(L^p\)-norm for \(p \geq 2\)) towards the solution of (1) and that its rate of convergence is of \(\frac{1}{2}\). The proof of this result is partially technical but makes several uses of Hölder, Burkholder-Davis-Gandy and Gronwall inequalities.NEWLINENEWLINEThe last part of the work is devoted to the investigation of the existence and the uniqueness of the solution of (2) and to the computation of the rate of convergence of the EM process associated with (2) towards the solution of (2). The existence and uniqueness of the solution is shown under the assumptions that \(b\) and \(h\) are Lipschitz with respect to the first argument and possibly highly nonlinear (including once again functions of polynomial growth of any order) with respect to the delay variable. Under the same assumptions on \(b\) and \(h\), the authors prove that the EM process converges strongly towards the solution of (2) with a rate close to \(\frac{1}{2}\) when choosing the quadratic norm (\(p=2\)). Except the technical and not obvious steps of the proofs, it uses Hölder and Gronwall inequalities to obtain the desired results.